Lumped coefficients

HO-oxidation of an individual NMHCj produces H02 radicals with a yield aj, and oxidation of the NMHC oxidation product produces H02 in stoichiometric amount The lumped coefficients or yields a and p need not be integers, and represent the effectiveness of a particular NMHCj in producing RO2. and H02 radicals (lumped together as HO2) that will then oxidize NO. to N02 in processes such as R6 and R13, producing one net ozone molecule each. Alternatively, when the NO. concentration is low, peroxyl radicals may form PAN (as in R22) or hydrogen peroxide (as in R33) which are other oxidant species. In this formulation, transport is expressed by an overall dilution rate of the polluted air mass into unpolluted air with a rate constant (units = reciprocal time dilution lifetime=1// ). This rate constant includes scavenging processes such as precipitation removal as well as mixing with clean air. 

Another approach to scale-up is the use of simplified models with key parameters or lumped coefficients found by experiments in large beds. For example, May (1959) used a large scale cold reactor model during the scale-up of the fluid hydroforming process. When using the large cold models, one must be sure that the cold model properly simulates the hydrodynamics of the real process which operates at elevated pressure and temperature. 

Written in this way, each global kinetic constant can be broken down into a sum of generally four terms. Each term is itself the product of intrinsic elementary constants—in common factor—and a (double) sum of contributions usually known as Lumping Coefficient —in white in Fig. 26—see also Section VI.B.l. There are as many lumping coefficients as there are different... 

These lumping coefficients contain no kinetic parameters and can be calculated independently using ... 

For the hydrocracking process, we must therefore find a way of directly calculating the lumping coefficients between chemical families (see Section V ), without having to generate the reaction network (no storage of molecules and reactions), Fig. 32. 

The methodology applied is based on factorisation of lumping coefficients into several elements which can be calculated independently ... 

A first Section (VI.B.l) describes the factorisation of the equation giving the expression of lumping coefficients,... 

The equation of lumping coefficients is the product of a sum of inverse symmetry numbers and an entire series of thermodynamic terms (see Section V.E). This complex sum is generally calculated by adding its component terms after generating a reaction network. If we examine the problem which led to this complex sum from a different angle, we can determine another equation which is as rigorous but formally simpler. 

Original and fast calculation method for lumping coefficients... 

On the basis of the simplified (in the mathematical meaning) equation of Fig. 33, we can identify a calculation logic all lumping coefficients can be broken down into a product of two terms describing the respective free enthalpies of the reactive paraffins and activated complexes involved.22 These two terms are calculated irrespective of the number of carbon atoms and the number of branches. The calculation is performed using recursive series and is therefore extremely fast (approximately two minutes for a C30 network limited to eight branches) this is discussed in Paragraph VI.B.2 below. 

We now need to calculate the sum of the reciprocals of the symmetry numbers of the activated complexes for all reactions included within a given lumping coefficient. This calculation requires two calculation principles ... 

By definition, homologous activated complexes have the same number of carbon atoms and the same number of branches. They are therefore involved in the same lumping coefficient. Consequently, each lumping coefficient has a sum of sums of the reciprocals of the symmetry numbers of each class of homologous activated complexes as a factor—Fig. 35 as cracking example. 

Total sorption is the sum of dissolution (D) and a hole-filling S U) components (eq 1). Sorption in the dissolution domain is given by a linear term where C is the fluid-phase concentration and is a lumped coefficient representing all available dissolution regions j (i.e., Kd = ZKdj ) Sorption in the hole-filling domain... 

The curves calculated in this way at constant for different combinations of Pe and S [i.e., for different combinations of axial dispersion and mass transfer resistance but the same linear combination 1/Pe + 5(1 + 5/0/15] show close agreement with each other and with the curve calculated from the simple linearized rate model using an overall lumped coefficient to account for the combined effects of axial dispersion, diffusion, and external mass transfer resistance. 

STRESS. Applies the variational recovery method to calculate nodal values of pressure and, components of the stress. A mass lumping routine is called by STRESS to diagonalize the coefficient matrix in the equations to eliminate the... 

Lujuid-Pha.se Transfer. It is difficult to measure transfer coefficients separately from the effective interfacial area thus data is usually correlated in a lumped form, eg, as k a or as These parameters are measured for the Hquid film by absorption or desorption of sparingly soluble gases such as O2 or CO2 in water. The Hquid film resistance is completely controlling in such cases, and kjji may be estimated as since x (Fig. 4). This... 

The rate law draws attention to the role of component concentrations. AH other influences are lumped into coefficients called reaction rate constants. The are not supposed to change as concentrations change during the course of the reaction. Although are referred to as rate constants, they change with temperature, solvent, and other reaction conditions, even if the form of the rate law remains the same. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

If the degree of superheat is large, it will be necessary to divide the temperature profile into sections and determine the mean temperature difference and heat-transfer coefficient separately for each section. If the tube wall temperature is below the dew point of the vapour, liquid will condense directly from the vapour on to the tubes. In these circumstances it has been found that the heat-transfer coefficient in the superheating section is close to the value for condensation and can be taken as the same. So, where the amount of superheating is not too excessive, say less than 25 per cent of the latent heat load, and the outlet coolant temperature is well below the vapour dew point, the sensible heat load for desuperheating can be lumped with the latent heat load. The total heat-transfer area required can then be calculated using a mean temperature difference based on the saturation temperature (not the superheat temperature) and the estimated condensate film heat-transfer coefficient. 

The geometry of the tubes allows the heat transfer being considered one dimensional, and each tube to be a lumped system in front of the ambient air. This two conditions are fulfilled when Bi < 0.1 (Biot number Bi = a /(/(2a ), where R is the radius of the sample, X its thermal conductivity and a the heat transfer coefficient between the tube and the environment). Once the temperature-time curves of the PCM and the reference substance are obtained (Figure 160), the data can be used to determine the thermophysical properties of the PCM. 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

In whole tissue or cell monolayer experiments, transcellular membrane resistance (Rm = Pm1) lumps mucosal to serosal compartment elements in series with aqueous resistance (R = P ). The operational definition of Lm depends on the experimental procedure for solute transport measurement (see Section VII), but its magnitude can be considered relatively constant within any given experimental system. Since the Kp range dwarfs the range of Dm, solute differences in partition coefficient dominate solute differences in transcellular membrane transport. The lumped precellular resistance and lumped membrane resistance add in series to define an effective resistance to solute transport ... 

The rate expressions have been written in generalized fashion with the terms fp, fb, fst, and fgt containing the reaction rate constants, stoichiometric coefficients, and concentrations of the various stable species present in the reaction mixture. If one also wished to consider bi-molecular radical processes, these could also be lumped into the / parameters. 

Similarly, for the frictional resistive force, Ff, lumping the geometry factors into a constant, c, and letting a = friction coefficient ... 

The main problem with 1-radical models is that they are chemically unrealistic. The distinction between oxidizing and reducing species is not recognized, and different rate and diffusion coefficients must be lumped into one. 

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